APPENDIX: TRANSMISSION LINE MODELLING AND PORT-BASED CIRCUITS

Transcription

1 APPENDIX: TRANSMISSION LINE MODELLING AND PORT-BASED CIRCUITS A. MODELLING TRANSMISSION LINES THROUGH CIRCUITS In Chapter 5 we introduced the so-called basic rule for modelling circuital systems through circuits, which required us to neglect (a) conduction and displacement currents through wires and (b) EMF induced by the effects of Faraday s law in the circuital systems space between wires. There are cases, however, in which these cannot be neglected. The most important is the example of very long high-voltage power lines that transmit power over a distance of many kilometres, for which the cumulative effects of displacement currents between conductors and of the self-induced voltages can be significant in line operation. In other cases, such as medium-voltage lines, it could be advisable to take into account the effects of self-induction while still neglecting displacement currents between conductors. In this appendix we show that if self-induction cannot be disregarded, the very concept of circuit becomes loose, and special constraints must be introduced to continue to use the usual circuit analysis techniques, and in particular Kirchhoff s laws. We do this by taking as reference a two-wire transmission line, the most important case for theoretical and practical reasons. Fundamentals of Electric Power Engineering: From Electromagnetics to Power Systems, First Edition. Massimo Ceraolo and Davide Poli. 4 by The Institute of Electrical and Electronics Engineers, Inc. Published 4 by John Wiley & Sons, Inc. 5

2 5 APPENDIX: TRANSMISSION LINE MODELLING AND PORT-BASED CIRCUITS (a) t i (t) t (b) t t t t t t t t t t (c) t t t t FIGURE A.. A physical system containing a transmission line (a) and a lumped components model candidate (b and c). A.. Issues and Solutions When Displacement Currents are Neglected Consider the physical system depicted in Figure A.a, for which we postulate that it is possible to build a voltage source that is, a device able to generate a time-varying voltage on its left side. This is known to be true from common engineering practice, and it should be part of the student s previous knowledge. The remainder of the circuit is constituted by a two-conductor line (let these conductors be two copper cylinders), connecting the voltage source with the load which is another cylindrical conductor possibly made of a different conducting material whose resistivity is higher than that of the line. Although simple, this is a distributed-parameter physical system, governed by electromagnetics equations. Can this system be modelled by a circuit of the type shown in Figures A.b and A.c, where the four conductive circuit elements are substituted by branches? If we look now at Figure A.b, terminals are set exactly at the borders of lumped components and are connected to each other by ideal wires, as graphically shown. Once this is clear, the terminal representation is normally omitted, and the circuit is written as in Figure A.c, where terminals are not shown and no distinction is made between the points of ideal wires, since all these points share the same current and the same potential. The time variability of electrical quantities changes things radically with respect to the DC case: Faraday s law applies and must be taken into account. Current flows not only within wires, but also between the upper and lower wire. In fact, the two wires, along with the insulating material between them, behave

3 MODELLING TRANSMISSION LINES THROUGH CIRCUITS 5 as a capacitor, the dielectric being the insulating material. Therefore there are displacement currents between them. The practical experience of electrical engineers has highlighted the fact that it is possible to analyse transmission lines which are not very long (i.e., no longer than 5 8 km), operating at 5 or 6 Hz, while neglecting the effects of conduction and displacement currents between the wires. For now, then, we will disregard the effects of these wire-to-wire currents; they will be considered at a later stage. As a consequence of this hypothesis, the current flowing inside the wires remains the same, from left-end terminals to right-end ones. The first question to answer for the circuital system of Figure A.a, is whether it is possible to define a potential function at the four terminals t, t, t, and t, if selfinduction EMF cannot be neglected. If the answer is yes, KVL applies and a circuit can be used to effectively model the system. The existence of a potential function for the circuital system of Figure A.a is equivalent to stating that the voltage across two of the terminals does not depend on the path used for measuring it. Therefore, we can imagine installing an ideal voltmeter in the circuital system and evaluating whether the voltmeter reading depends on its position. A voltmeter is a device which is able to show the difference of potential at the two points of the circuital system to which it is connected. An ideal voltmeter is a device that draws no current (or an infinitesimal current) when it is inserted into the circuit and therefore its insertion does not alter the previous equilibrium of the system. Consider the two ideal voltmeters V, and V, and the corresponding measuring wires depicted in Figure A.. Because the voltmeters are ideal, the currents flowing in the wires connecting t and t with the voltmeters are negligible with respect to the t i (t) t A u V H l tt t t i (t) t A t H l tt u V t L l FIGURE A.. Experimental evaluation of u of the system shown in Figure A..

4 54 APPENDIX: TRANSMISSION LINE MODELLING AND PORT-BASED CIRCUITS main circuit current i(t). The following equations can be written, combining Faraday s law and Ohm s law for the shaded areas A, and A : A : u t R it d dt ψ At A : u t R it d dt ψ At (A.) where R is the resistance of the system s upper wire between t and t. ψ A, ψ A fluxes are linked with surfaces of areas A and A, respectively. Undoubtedly, voltages u and u measured by voltmeters V and V, respectively, are different from each other, because the two flux linkages ψ A and ψ A are different, being caused by the same electric field due to i(t) but integrated in two different areas A and A A. Therefore, since the quantity measured as voltage between two points depends on the path in which we install the voltmeters, it is impossible to define the potentials of terminals t and t. Since the existence of potential in all the connection points (element terminals) implies the applicability of the KVL and vice versa, KVL itself cannot be applied in the standard way, and therefore no standard circuit can be created for the system containing the transmission line. These results can be repeated whenever it is not deemed possible in a circuital system to neglect self-induction EMFs in the space between the wires that connect the components of the system component. The following can thus be stated: Result: Circuit modelling of transmission lines In a circuital system, when the self-induction electromotive force cannot be neglected in the meshes created by lines connecting devices, a potential function of all the points of the system cannot be defined, and therefore no standard circuit can be defined to model the system. A system containing a long transmission line, however, in which the length is much greater than the distance between the conductors, can be analysed. Consider Fig. A., in which L l H l. In this case, the following equations can be written as d u t et R c it dt ψ A A t u t et R c it d dt ψ A A t where R c is the sum of upper and lower conductor resistance. (A.)

5 MODELLING TRANSMISSION LINES THROUGH CIRCUITS 55 t i(t) t tt A A u V t t i(t) t A A u H l t t V L l t FIGURE A.. Experimental evaluation of u of the system shown in Fig. A.. Since it is assumed L H, it can also be said that A A and A A, and therefore the flux ψ A A linked with area A A and the flux ψ A A linked with A A can be assumed to be equal to each other, as follows: d dt ψ A A t d dt ψ A A t d dt ψ A t d dt Lit L dit dt where L is the self-inductance coefficient of the line and is related to the area A. The two equations (A.) can therefore be collapsed into a single equation: ut et R c it L dit dt (A.) Equation (A.) can be the equation describing the electrical behaviour of the metacircuit shown in Figure A.4. The word metacircuit and the dashed lines indicate that this is not a true circuit since it cannot be used to compute voltage differences between a terminal existing at the sending end of the transmission line and one existing at the receiving end, but only voltage differences between terminals at the same end of the line. This metacircuit concept, though important, is usually ignored in circuit and electrical engineering textbooks. Nevertheless, any electrical engineer knows that when long transmission lines are involved, it is not correct to evaluate voltage differences between terminals situated at two opposite ends of the line. To our knowledge, a rational analysis of why this The word metacircuit was invented here to make it clear that the drawing shown is not actually a circuit, since not all potential differences between terminals have physical meaning and can be computed.

6 56 APPENDIX: TRANSMISSION LINE MODELLING AND PORT-BASED CIRCUITS i(t) R c L R u FIGURE A.4. A metacircuit representing the circuital system depicted in Figure A.. happens has not been previously published, at least in the more widespread literature; the contribution of this appendix is thus significant. In the next section the analysis is repeated while also taking displacement currents into account. A.. Steady-State Analysis Considering Displacement Currents Consider again a two-wire transmission line (Figure A.5), this time taking into account the displacement current. Consider in particular a single line segment of length dx (the shaded area in the figure). This segment will have an infinitesimal inductance dl Ldx if L indicates the inductance per unit length, and it will have an infinitesimal resistance dr Rdx if R indicates the resistance per unit length. The induction phenomenon along with the ohmic drop can be taken into account by writing the following equation: ux dl di dx dr? i ux dx C i i(x) u(x) i(xdx) u(xdx) u di = L R i x dt i du = C G u x dt dx x l FIGURE A.5. Two-wire transmission line equations.

7 MODELLING TRANSMISSION LINES THROUGH CIRCUITS 57 or: ux L dx di dx R dx? dx then L di dx In a similar way, conduction and displacement currents across the two wires in a segment of length dx can be found by resorting to conductance per unit length G and capacitance per unit length G: ix C dx du dx G dx? u ix dx then C du dx These developments justify the equations in Figure A.5. If we consider the line operating in steady state at the angular frequency ω, the same equations can be written as follows (uppercase underlined quantities indicate the complex numbers that represent jωl? IxR Ix jωc? UxG Ux Y l Ux (A.4) Equation (A.4) can be easily solved by the steps described in all books dealing with transmission lines and omitted here for simplicity; the following are given instead: U x U? cosh Kx Ix U Z c Z c I? sinh Kx? sinh Kx I? cosh Kx (A.5) where Z c qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R jωl =G jωc and K qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R jωl? G jωc This is the capacitance per unit length of the capacitor, whose conductor plates, which are parallel to each other, are the two wires and the dielectric is the air between them.

8 58 APPENDIX: TRANSMISSION LINE MODELLING AND PORT-BASED CIRCUITS I () I (l) Z I () Z Z I (l) U () Y Y U (l) U () Y U (l) FIGURE A.6. Circuits candidates to model the system of Figure A.5 in steady state. Suppose that now it is of interest to know only the behaviour of the line at the two ends: sending end (terminals ) and receiving end (terminals ). From equation (A.5) it follows that U l D? U B? I; U A? UlB? Il I l C? UA? I; I C? UlD? Il (A.6) where A D cosh Kl; B Z c sinh Kl; C sinh Kl=Z c note that AD BC As for the simpler case above, consider now the circuits depicted in Figure A.6. Z B; Y Y A =B Z Z A =C It is easy to verify that equations (A.6) are valid for both circuits. However, it would be erroneous to state that the circuits in Figure A.6 are steady-state equivalents of the transmission line: those circuits, indeed, are branch-based circuits and, as such, allow the determination of quantities such as voltage U AC, something which is impossible (because it has no physical meaning) through the equations of the physical line. Therefore, the obtained steady-state line model can again (see Section A..) be better represented graphically using the metacircuit concept (Figure A.7) I () Z I (l) I () Z Z I (l) U () Y Y U (l) U () Y U (l) FIGURE A.7. Metacircuits representing a transmission line operating in steady state.

9 MODELLING TRANSMISSION LINES THROUGH CIRCUITS 59 More in Depth: Line models with circuits containing : transformers One could be tempted to avoid the introduction of the metacircuit concept for transmission lines operating in steady state and instead introduce a : ideal transformer at the beginning or the end of the models as shown in the case of the PI line model in the upper part of the diagram below. This, however, would be tricky and is better avoided, since the use of this line equivalent would not prevent cross-impedances being added, such as the Z new shown in the lower part of the diagram below. This is indeed erroneous because analysis of this network using Kirchhoff s equations would lead to results that do not correspond to the system physics. I () : Z I (l) I () Z : I (l) U () Y Y U (l) U () Y Y U (l) line equivalent line equivalent I () Z Z new : I (l) U () Y Y U (l) A.. Practical Considerations For low-voltage and medium-voltage lines (up to 5 kv nominal), considering the actual parameters (inductances per unit length and capacitances per unit length) and the limited lengths, the admittances Y can be neglected and the line can be represented by a metacircuit containing only the longitudinal impedance Z or Z Z. Moreover, for low-voltage lines ( kv nominal and below), for conductor sections up to 95 mm, the longitudinal reactance can also be neglected with respect to the resistance. For those lines, by and large the most common in domestic and civil environments, lines are modelled just by an equivalent resistor.

10 5 APPENDIX: TRANSMISSION LINE MODELLING AND PORT-BASED CIRCUITS A more formal approach showing how to use modified circuits in steady state to analyse systems in which displacement currents between wires and self-induction voltages cannot be neglected is illustrated in the next section. In high-voltage lines and very high voltage lines (e.g., in the range 8 kv) the Y component of the model can be neglected only for short line trunks, usually up to km. A. MODELLING LINES AS TWO-PORT COMPONENTS A.. Port-Based Circuits In the previous section it was shown that transmission lines, which are very important and frequently used components, do not fit easily with the circuit concept used in this book (and introduced in Chapter ) in either their general or branch-based form. This is mainly because, differently from circuits, the mathematical line model should reflect the fact that a voltage across two terminals at the sending end and receiving end of the line is undefined. Therefore, so as not to miss out on the considerable advantages of the circuit-based technique in analysing physical systems with a circuital nature (devices connected by wires), a new definition of circuits must be introduced. Consider the components shown in Figure A.8. Consider first the upper part of the figure. The 4-terminal component of Figure A.8a, by its very constitution, places constraints on the terminal s electrical voltages and currents. The independent voltages (differences between terminal potentials) and currents amount to six quantities, since the fourth current is determined by Kirchhoff s Current Law (KCL), and all potential differences can be determined as differences between the three potentials between terminals to and terminal. The component shown in Figure A.8b still has four terminals, but these are grouped into two ports. Something similar happens with Figures A.8c and A.8a, where more terminals are involved, and each port contains m terminals instead of two. The definition of the two-port m component outlined here is as follows : A two-port m component is an n-terminal component with the following special features:. The n terminals, which must be even in number, are grouped into two groups of m n/ terminals, called ports.. It is postulated that the sum of all currents entering any port is always zero.. When the two-port component is inserted into a circuit, the circuit must never be used to determine voltage differences between potentials of terminals belonging to different ports, since they are meaningless. This is a more structured definition of two-port components that can be found in textbooks (see Chapter 9 of [bc]). The expression two-port m component is equivalent to component with two m-terminal ports.

11 MODELLING LINES AS TWO-PORT COMPONENTS 5 (a) i u (b) i i i u 4-terminal component i u i -port u u component i i i i (c) (d) i i i n (t) u n u i n-terminal component (n = m) u i u i (t) i k u u i i u k -port m component i i...i n i i...i m i i...i m FIGURE A.8. From 4-terminal and n-terminal components to -port components. u k u u i k Items and mean that KCL and Kirchhoff s Voltage Law (KVL) must be valid separately for each port (and each circuit section connected to that port) but not for the circuit as a whole. A.. Port-Based Circuit and Transmission Lines The two-port component is a suitable tool for modelling a circuit containing a transmission line such as the one depicted in Figure A.5. When subsystems are connected to the left end and right end of the line, the whole system can be modelled by connecting the branch-based circuit modelling the lefthand subsystem on the left, the two-port component modelling the line, and a branchbased circuit modelling the right-hand subsystem on the right. The situation is summarized in Figure A.9. The circuital system containing the line is recalled in Figure A.9a for simplicity considering very simple subsystems at the left and right end of the line. The line can be effectively modelled as a two-port component as in Figure A.9b. Should conductive and displacement currents between the wires be disregarded, the two-port component behaviour is described by the single equation in Figure A.9c, while in the more general case in which these currents between wires are taken into consideration, in steady state, the line is described by a two-port component with the equations in Figure A.9d. The values of A; B; C; D are those immediately given below equation (A.6). Figures A.9e and A.9f show the metacircuit counterparts of the port-based representations of Figures A.9c and A.9d. The metacircuit approach is equivalent

12 5 APPENDIX: TRANSMISSION LINE MODELLING AND PORT-BASED CIRCUITS (a) i i(x) i (b) u u R u two-port comp. modelling the line R u i i (c) i E di u u u u = Ri L R dt u i (d) I I U U = A U B I I = C U D I U R u (e) i(t) R L (f) I I a E u u R u U Y Y U R u FIGURE A.9. The two-wire transmission line and related different circuital representations. to the port-based one, but port-based components are more usually represented by their equations, rather than by circuit-like drawings. In normal engineering usage, these curveddashed lines are omitted. Normally this does not create problems, since the engineer retains a visual memory of them and will avoid the mistake of evaluating cross voltages between the ports, even when they are not explicitly shown. A.. A Sample Application Transmission lines are used largely to transmit electric power at distances that range from some metres (household installations) to hundreds of kilometres (high-voltage three-phase transmission lines). A significant example of a three-phase transmission system is shown in Figure A.. It is composed of a generator (electric machine that generates electric power from mechanical power), some transformers, two three-phase electric lines, composed of three wires (e.g., the overhead conductors of high voltage lines) and the ground below them, and, finally, the distribution network, which acts as a load for the rest of the system shown. In the figure, transformer transfers electrical energy from the medium-voltage level to the high-voltage level and transformer does the opposite.

13 FINAL COMMENTS 5 MV HV MV Line Line Gen Transf Transf FIGURE A.. Example of a power system. Distribution Network p p Gen Transf Line Transf Line Distrib. net Port Port FIGURE A.. Circuital representation of the power system of Figure A.. As usual in power system analysis, only a one-wire equivalent is shown. Indeed all apparatuses shown (generator, transformer, etc.) have four-terminal ports: the three phases plus an additional terminal which is normally connected to the ground. Consequently, the power system depicted in Figure A. can be represented as in Figure A.. The different apparatuses are either 4-terminal components (generator, distribution network), or two-port 4 components (lines, transformers). The behaviour of each apparatus is described by its constitutive equation, similar to those given for the two-wire transmission line in Figure A.9, but more complex because of the higher number of terminals per port. The connections between the port-based components shown in Figure A. indicate the equality of voltage of terminals connected to each other, as well as the equality of the current in the shown wires, paying attention to their signs. This representation does not allow evaluation of the voltage difference between terminals belonging to two different ports of any two-port components, for example, U U p U p. This does not create limitations in the applications, because ports and, and therefore the terminals p and p, might be as much as several hundred kilometres from each other. A. FINAL COMMENTS This appendix has described a way to model transmission lines in circuit analysis even in cases in which it is not possible to neglect self-induced voltages (Faraday s law) and/or displacement currents between wires. The approach outlined here is useful, and commonly adopted, in the analysis of very high voltage transmission networks.

14 54 APPENDIX: TRANSMISSION LINE MODELLING AND PORT-BASED CIRCUITS The analysis presented in this appendix, however, gives only basic information geared to making the problem and the approach clear to students, but it is not sufficiently in-depth to give them adequate know-how for dealing with three-phase transmission systems containing long lines in a quantitative way.